QUESTION $1$

Solve the given first$-$order linear equation

$4ydx-(3\sqrt[\phantom{2}]{y}-2x)dy=0$

$[15]$QUESTION $1$

Solve the given first$-$order linear equation

$4ydx-(3\sqrt[\phantom{2}]{y}-2x)dy=0$

$[15]$

QUESTION $1$

Solve the given first$-$order linear equation

$4ydx-(3\sqrt[\phantom{2}]{y}-2x)dy=0$

$[15]$

QUESTION $2$

Use an appropriate substitution to solve

$(y+\sqrt[\phantom{2}]{x^2-y^2})dx-xdy=0$

QUESTION $3$

Solve the initial$-$value problem

$x^2\frac{dy}{dx}=y-xy,y(-1)=-1$

QUESTION $4$

Given that the function $y(x)=x$ is a solution of the corresponding homogeneous equation of

$x^2\frac{d^{2}y}{d{x}^2}-(x^2+2x)\frac{dy}{dx}+(x+2)y=x^3{e}^x$

use it to obtain the general solution of the nonhomogeneous equation.

$[20]$

QUESTION $5$

Use the method of the Laplace transform to solve the initial$-$value problem

$4y^{\text{'}\text{'}}+12y^{\text{'}}+9y=36t^3+256,y(0)=1,y^{\text{'}}(0)=\frac{63}{2}$

$[15]$

QUESTION $6$

When a mass of $2$ kilograms is attached to a spring whose constant is $32\frac{N}{m},$ it comes to rest in the equilibrium position. Starting at $t=0,$ a force equal to $f(t)=68e^{-2t}cos4t$ is applied to the system. Find the equation of motion in the absence of damping.

$[20]$

TOTAL MARKS: $[100]$